and
Γf :V ×R→V ×R×Rk(x, λ)7→(x, λ, g1(x, λ), . . . , gk(x, λ)) (again by abuse of notation) by
θ:V ×Rk →V (x, t)7→X
i
tiFi(x) and
Γf :V ×R→V ×Rk(x, λ)7→(x, g1(x, λ), . . . , gk(x, λ)).
If we set Σ =θ−1(0) for our new definition of θ, then f tG 0 ∈V at (0,0) ∈ V ×Rstill holds iff ΓftΣ at (0,0).
Now, consider the map
γf :V ×R→Rk
(x, λ)7→(g1(x, λ), . . . , gk(x, λ)).
Note that Γf(x) = (x, γf(x)) is the graph ofγf.
Moreover the canonical stratificationS of Σ⊂V×RK yields a stratification of Rk = ΣG = Σ(G): For each isotropy type τ, set Sτ = ∪s∈SSτ. By [Fie07, Theorem 6.10.1],Sτ is a Whitney stratification of Στ consisting of S-strata.
Thus,f tG0∈V at (0,0) is equivalent toγf tS(G). The next lemma gives some insight into the stratificationS(G). From now on, we fix a minimal set of homogeneous polynomial generatorsF1, . . . , Fk withF1(x) =x. ThenFk is of order≥2 forx≥2.
Lemma 6.26. Every stratumS⊂ S(G)withcodimS >0 inRk is contained in Rk−1:=
t∈Rk
t1= 0 .
Proof. Consider a point (0, t) ∈ Rk = {0} ×Rk ⊂ V ×Rk. Since θ(x, t) = P
itiFi(x), we obtain dxθ(0, t) =t11. If t1 6= 0, the implicit function theorem yields that there is a neighbourhood U of (0, t) whose intersection with the set Σ = θ−1(0)⊂ V ×Rk coincides with U ∩Rk. By the construction of the canonical stratification,U∩Rk is contained in a single stratum ofS.
This alternative characterization of G-transversality has some advantages:
• For a fixed choice of a minimal set of homogeneous polynomial generators F1, . . . , Fk, the mapγf is independent of the choice of the representation f(x, λ) = P
igi(x, λ)Fi(x) (see [Fie07, Lemma 6.6.3] and the definition of γafter his Corollary 6.6.1). This yields an elegant alternative to Bier-stone’s proof of the independence of the choice of the representation. As in Bierstone’s proof, a generalG-representationV is considered as a product V ×Rs, whereVG={0}andGacts trivially on the parameter spaceRs.
• We would like to prove openness and density of the propertyf tG 0∈V at (0,0) within the function space that we choose for the investigation of bifurcation problems. Openness follows from the openness in the space
CG∞(V ×R,R). For the proof of density, we give another equivalent de-scription ofG-transversality:
Consider the map γ:f 7→γf fromCG∞(V ×R, V)∗ or V0 to C∞(R,Rk).
γ is continuous ([Fie07, Lemma 6.6.7]). The image is given by the set of functions whose first coordinate is given byσ(λ) orλrespectively. Thus, γf(0)∈Rk−1 and γf is transverse to Rk−1 at 0. By Lemma 6.26, inter-secting the strata ofS(G) withRk−1 yields a Whitney stratificationAof Rk−1 andγf is transverse to S(G)ifγf(0) is contained in a stratum ofA of codimension 0.
Since this holds for a dense subset of the image ofγ, the preimage of this set inCG∞(V ×R,R)∗or V0 is also dense.
• This alternative description ofG-transversality confirms that a vector field f ∈CG∞(V×R,R)∗isG-transverse to 0 at (0,0) if this is true for thed-jet off0, where dis the maximal degree of the Fi. We will see below that in this case the homeomorphism class of the zeros off is stable under pertur-bations. This yields Field’s finite (weak) determinacy result. (In Field’s terminology, broadly speaking, a representation is weakly d-determined if for a generic equivariant smooth family fλ, the topological properties of the zero set are stable under perturbations and determined by jdf0. Determinacy requires also hyperbolicity of the non-trivial zeros.)
If f ∈CG∞(V ×R,R)∗ is G-transverse to 0 at (0,0), the local zero set off is a Whitney stratified subset ofV ×Rwhose structure may be deduced from that of Σ.
The following smoothness result for the isotropy components Στis essential:
For any isotopy typeτ = (H), set gτ := dimG
H = dimG−dimH, nτ := dimN(H).
H = dimN(H)−dimH.
Lemma 6.27 ([Fie07, Lemma 6.9.2]). For each isotropy type τ of V, the set Στ is a smooth manifold with
dim Στ =k+gτ−nτ.
If f tG 0 at (0,0), Γf is transverse to the canonical stratification S of Σ along a neighbourhoodUof (0,0). Since for eachτ, the stratificationSτconsists ofS-strata, Γf is transverse toSτ and hence to Στ alongU as well. Thus the zeros off of isotropy typeτ contained inN form a dimgτ−nτ+ 1-dimensional smooth manifold ofV ×R, whose closure contains the origin (0,0) if they exist.
We first consider the case of a finite group G: In this case, we obtain 1-di-mensional smooth manifolds, whose boundaries consist of the origin. Each of these curves is Whitney regular over the origin. Let us call each union of the origin and one of these curves abranch. Applying a result of Pawlucki on regu-larity of Whitney stratified semi-algebraic sets to Στ and a stratum in ¯Στ∩Rk, Field and Richardson even show that each branch may be parameterized as a C1-curve [0, δ)→V ×R.
If dimG >0, the zeros of isotropy type τ consist ofG-orbits of dimension gτ. Thus, if nτ >1, we do not expect any zeros of isotropy type τ near 0. If
nτ = 1, connected components of zeros of isotropy typeτ inN consist of single G-orbits. Since the zero set off is Whitney stratified, by local finiteness, they are bounded away from zero. Hence in the local sense, there are no zeros of isotropy type τ near zero. Only in the case nτ = 0, we obtain a “branch” of G-orbits of zeros: Ifp1, . . . , plform a minimal set of homogeneous generators of the ringP(V)G, the orbit mapP×1= (p1, . . . , pl, λ) maps each of these orbits to a single point ofRl×R. As in the finite case, it may be shown that the image inRl may be expressed as a union ofC1-curves starting at (0,0).
For a non-finite group G, it seems to be more natural to include relative equilibria. This is illustrated by the above results: Local branches of zeros of isotropy type τ only occur in the case nτ = 0. In this case, all relative equilibria are in fact equilibria, since the trajectory of a relative equilibrium with isotropy subgroupH is contained in itsN(H)-orbit. Moreover, in the case nτ = 1, isolated orbits of zeros may occur. As examples show, these are usually embedded in branches of relative equilibria. The analysis of the local structure of relative equilibria yields that this is indeed the behaviour we generically expect.
The generalization of the results for finite groups to the bifurcation of relative equilibria first appeared in [Fie96]. For the generalization, we have to consider also bifurcations that are caused by a pair of purely imaginary eigenvalues±αi, α >0. Centre manifold reduction or a Lyapunov-Schmidt reduction in a way used for the proof of Hopf bifurcation theorems yields an equivariant familyfλ
of vector fields on the real part V of the sum of the generalized eigenspaces for±αi. Generically, the generalized eigenspaces coincide with the eigenspaces such that df0(0)2 =−α21. Thus, 1αdf0(0)2 defines a complex structure onV. Moreover, it may be shown thatV with respect to this complex structure forms a complex irreducible G-representation. Using the theory of Birkhoff normal forms, we may assume that the Taylor polynomial Trf of order r commutes with the S1-action defined by the complex structure for an arbitrary finiter.
Field proves that forr large enough, the branches of relative equilibria ofTrf persist if the higher order terms of f are added. (But there may be additional relative equilibria forf.) Thus, Field restricts his analysis to the case of complex irreducible representations of compact groups of the formG=K×S1. For these, he considers the set of normalized families
V0(V, G) ={f ∈CG∞(V ×R, V)|dfλ(0) = (λ+ i)1}.
(The normalization consists of a scaling of time which corresponds to a scaling of f and a reparameterization in the variable λ afterwards.) The rest of the argument is similar to the one for bifurcations of equilibria:
The set Σ is replaced by the set
Σ∗ ={(x, t)|θ(x, t)∈TxGx}.
Note that Σ∗ is an algebraic set, since (x, t)∈Σ∗ iff dP(x)θ(x, t) = 0:
Suppose that x has isotropy type τ. Since θ(·, t) is an equivariant vector field,θ(x, t)∈VGx⊂TxVτ.
For every connected component Vτi ⊂ Vτ, the image P(Vτi) = Vτi. G is a smooth manifold and P : Vτi → Vτi.
G is a submersion (see [DK00, Re-mark 2.7.5]). Hence forx∈Vτ, the kernel of dP(x)
T
xVτ coincides withTxGx.
The isotropy components of Σ∗ also form smooth manifolds:
Lemma 6.28 ([Fie07, Lemma 10.2.2]). For each isotropy type τ of V, the set Σ∗τ is a smooth manifold with
dim Σ∗τ =k+gτ.
As before, each Σ∗τ is a union of strata of the canonical stratification of Σ∗. Thus, for any f ∈ V0(V, G) with Γf t Σ∗ at (0,0), the set of pairs (x, λ) such thatx is a relative equilibrium offλ consists of branches of G-orbits (in the same sense as above).
It only remains to show that this transversality condition is open and dense in V0(V, G). Openness is clear. Density is proved in a similar way as above:
We choose a minimal set of homogeneous equivariant generators F1, . . . , Fk with F1(x) = x and F2(x) = ix. Again, Σ∗τ ∩Rk is contained in Rk−1 = t∈Rk
t1= 0 . Thus, again we obtain a Whitney stratificationA∗ of Rk−1 such that the transversality condition is satisfied if
γf(0) = (0,i, g3(0,0), . . . gk(0,0))
is contained in a stratum ofAof codimension 0. Moreover, using ix=TxS1x⊂ TxGx, it may be shown that the stratification A∗ is invariant under transla-tion in the directransla-tion of the t2-axis. As above, this yields the density of our transversality condition.
Remark 6.29. The transversality condition can be formulated in terms of equivariant transversality: Consider the algebraic subset
T :={(x, v)|dP(x)v= 0} ⊂V ×V and the map
1×θ:V ×Rk →V ×V (x, t)7→(x,X
i
tiFi(x)).
Then Σ∗ = (1×θ)−1(T). The space V ×V may be identified with the space J0(V, V) and1×θcoincides with the mapU0. The compositionU0◦Γf yields the graph of f that may be seen as the 0-jet of f. Thus, in the notation of equivariant 0-jet-transversality, the condition is j0f tG T. Equivariant 0-jet-transversality is just a slight generalization of equivariant 0-jet-transversality.
Field also considers an equivariant 1-jet-transversality condition in order to show that the bifurcating equilibria and relative equilibria are generically hyperbolic and normally hyperbolic respectively. For the characterization of normal hyperbolicity in this context, the splittingf =fT +fN of a vector field f ∈CG∞(V, V) within a tubular neighbourhood is used, wherefT is tangential to theG-orbits andfN is tangential to the slices. The existence of such a splitting is a result of Krupa [Kru90]. A G-orbit of relative equilibria is a normally hyperbolic submanifold forf iff each of its elements is a hyperbolic zero of fN
restricted to the corresponding slice. Equivalently, for any element x of the orbit and any equivariant vector field ˜f with ˜f(x) =f(x) that is tangential to theG-orbits, the centre space of d(f−f˜)(x) has dimension dimGx.
LetH(V)⊂EndR(V) be the semi-algebraic subset consisting of hyperbolic manifolds. In the case of a finite groupG, Field sets
Z1:=
(x,0, A)∈J1(V, V) =V ×V ×EndR(V)
A /∈H(V) .
For the non-finite case, the definition of a corresponding setZ1∗is more involved.
Starting with the semi-algebraic set
Z0∗(τ) ={(x, v)∈V ×V |x∈Vτ, v∈TxN(G)x}
for any isotropy type τ, a map Ξ from Z0∗(τ) into the set of vector fields CG∞(V, V) is constructed such that (Ξ(x, v)) is tangential to theG-orbits and (Ξ(x, v))(x) =v. Based on these maps, one obtains a semi-algebraic setZ1∗with the property that theG-orbit of relative equilibriumxis a normally hyperbolic submanifold forf iffj1f(x)∈/Z1∗.
In both cases, Field proves that for dimensional reasons, j1f tG Z1 and j1f tGZ1∗ are equivalent to j1f(x)∈/ Z1∗ andj1f(x)∈/Z1∗ respectively. Again, openness and density of this condition can also be shown for the setsV0(V, G).
These methods of analysing questions in bifurcation theory have strongly inspired the proceedings of this thesis.
As mentioned in the beginning, Birtea et al ([BPRT06]) also build on Field’s ideas to study bifurcations in 1-parameter families of equivariant vector fields. In particular, they investigate bifurcations of relative equilibria in the Hamiltonian case. Their analysis relies on a variant of Field’s method proposed by Kœnig and Chossat ([KC94]): For a given equivariant vector fieldX on a G-represen-tationV, we consider the projection ˜X to the orbit space V
G⊂Rk, given by X˜(P(x)) = dP(x)X(x). The space V
G is Whitney stratified, where the strata are given by the images of the subsets of the same isotropy type. The projection X˜ is a vector field on V
G, in the sense that it is tangent to the strata. If the familyf is represented byf(x, λ) =P
igi(P(x), λ)Fi(x) then we have f˜(P(x), λ) =X
i
gi(P(x), λ) ˜Fi(x).
Instead of the mapθ:V×Rk→V, we consider the induced map ˜θ: V
G×Rk → V
G given by
θ(P˜ (x), t) =X
i
tiF˜i(x).
Then ˜Σ := ˜θ−1(0) coincides with the projection of the set Σ∗ ⊂ V ×Rk to V
G×Rk. Moreover, the projections of the strata of Σ∗ form a stratification of ˜Σ. The induced stratifications ofRk are the same, when we identifyRk with the subsets{0} ×Rkof Σ∗and ˜Σ respectively. Kœnig and Chossat say that the projected family ˜f isG-transverse to 0 =P(0)∈ V
G at (0,0)∈ V
G×Riff the mapγwithγ(λ) = (g1(0, λ), . . . , gk(0, λ)) is transverse to this stratification ofRk. This is equivalent to Field’s condition.
Now, Birtea et al formulate a Hamiltonian analogue of equivariant transver-sality theory: They note that for aG-invariant Hamiltonian functionhwith a representationh=g◦P, the Hamiltonian vector fieldXh is given by
Xh(x) = Σli=1∂ig(P(x))J∇pi(x),
where ω = h·, J·i. Thus, if we redefine θ by θ(x, t) := Pl
i=1tiJ∇pi(x) for (x, t)∈V ×Rland Γh by Γh(x) := (x, ∂1g(P(x)), . . . , ∂lg(P(x))), thenXh can be decomposed asθ◦Γh. This is in principle what we will do in section 6.3 of this chapter. We will give our definition in terms of 1-jet-transversality in order to deduce directly from Bierstone’s theory that our transversality condition is well-defined and generic – a matter that Birtea et al take no notice of when they give their definition.
For a family f of Hamiltonian functions f(·, λ), we obtain an analogous decomposition. Alternatively, one could transfer Kœnig’s and Chossat’s formu-lation. This is the approach of Birtea et al ([BPRT06]). They observe that the projections of the vector fieldsJ∇pi are given by
{p1, pi} {p2, pi}
. . . {pl, pi}
.
Thus the map ˜θ: V
G×Rl→ V
G⊂Rlinduced byθis given by (x, t)7→A(x)t, where thel×l-matrixA(x) has the entries (A(x))ij ={pi, pj}(x). We also define Σ, and˜ S(G) in an analogous way as for general equivariant vector fields; i.e.
Σ := ˜˜ θ−1(0) and S(G) is the stratification ofRl ⊂Σ induced by the canonical˜ stratification of ˜Σ.
Birtea et al consider families of Hamiltonian functions parametrized byλ∈R such that at λ= 0 the derivative of the Hamiltonian vector field at the origin has either an eigenvalue 0 or a pair of non-zero purely imaginary eigenvalues±βi such thatE±βi is not irreducible as aG-symplectic representation. In the first case, generically this can be reduced to the study of familiesf :V ×R→R, where V is an irreducible symplectic representation and dXf(·,λ)(0) = σ(λ)J withσ(0) = 0 andσ0(0)6= 0. In the second case, the generic situation can be reduced to the case thatV is a sum of a pair of complex duals and dXf(·,λ)(0) is given by a sum of four matrices, each of which is a product of a constant matrix and a function inλ. One of these functions is also calledσand satisfies σ(0) = 0,σ0(0)6= 0. (This is a result from [COR02]. See [BPRT06] or [COR02]
for the precise form.)
Iff :V ×R→Ris such a family of Hamiltonian functions with Xf(·,λ)(x) = Σli=1∂ig(P(x), λ)J∇pi(x),
equivalently
X˜f(·,λ)(x) =A(x)∇P(x)g(P(x), λ),
Birtea et al set γ(λ) := (∂p1g(0, λ), . . . , ∂plg(0, λ)) ∈ Rl) and call the family X˜f(·,λ)transverse to 0∈V
G atλ= 0 iffγis transverse to S(G)at 0.
Then, they consider such families that satisfy this transversality condition and state that for a special class of symplectic representations V, the relative equilibria form ’branches’ as they do for generic equivariant families of general vector fields. That is, they locally formC1-curves in the orbit space V
G×R that start at (0,0) and contain no point (0, λ) withλ6= 0. The condition onV is, that there are indicesi0, i1∈ {1, . . . , l}, such that the function{pi0, pi1}does not vanish identically and there isx0∈V with limx→x0{p{pi0,pi}(x)
i0,pi1}(x) for alli6=I1
andxin the domain of definition. The proof, however, is erroneous: The authors
claim, that this condition implies that all strata ofS(G)of codimension≥1 are contained in the subspace{ti1 = 0} ⊂Rl and that it is hence possible to adapt Field’s argument. First, such an adaption of the arguments is only possible ifσ coincides with thei1-th entry of the functionγ. Second, the proof of the claim about the stratification relies on the assertion, that if thei0-component of map θ˜vanishes for some pair (P(x0), t0), then thei0-component of ˜θ(·, t0) vanishes identically. This is obviously not true.
Moreover, if there was aG-symplectic representationV such that the relative equilibria of a generic family of Hamiltonian systems with G-symmetry on V form branches of this kind, a single generic Hamiltonian function would have no non-trivial relative equilibria in some neighbourhood of the origin. This is not the behaviour that we expect. Indeed, as we have shown in chapter 3, if the connected component G◦ of the identity acts non-trivially on V, there is an open set of Hamiltonian functions hsuch thatG◦ acts non-trivially on the corresponding centre space Ec of dXh(0). In this case, the results discussed in chapter 5 show that we have to expect relative equilibria near 0 ∈ V. For example, this follows from remark 5.18.
Nevertheless, it is an interesting observation that ˜θtakes this particular sim-ple form. It may lead to a better understanding of the structure of Hamiltonian relative equilibria in symplectic representations in future work. The approach of the following sections, however, is different:
In section 6.3, we consider an algebraic set similar to that considered by Field for his investigation of the bifurcation of relative equilibria. We only need a suitable notion for equivariant transversality of Hamiltonian vector fields which will be given in terms of equivariant 1-jet-transversality. As mentioned, this is in principle equivalent to the definition given by Birtea et al ([BPRT06]).
The analysis of torus representations of section 6.4 is similar to Field’s orig-inal method of analysing the bifurcation of zero sets: As suggested by Chossat et al. in [CLOR03], we search for zeros of the augmented Hamiltonian and con-sider theξ∈gas parameter. Ifgis Abelian, the action on the parameter space is trivial and thus the problem can be handled as above. The only difference is the dimension of the parameter space, which is not an obstacle. As mentioned, more dimensional parameter spaces even occur in the proof of the independence of choices.